The radiation a difference in falling would make

The wave that stretches one way and squeezes the other gives the standard reason gravitational radiation begins at the quadrupole. A monopole would need the total mass to change, which it cannot. A dipole would need the centre of mass to accelerate, and momentum conservation forbids that. What is left is the quadrupole, the changing shape of the mass distribution, and it is feeble: suppressed by two more powers of the ratio of the orbital speed to the speed of light than a dipole would be.

That reason is correct and it hides an assumption. The dipole moment that would radiate is not the centre of mass. It is the centre of whatever the field couples to — the centre of gravitational charge — and the two coincide only because every body’s gravitational charge is exactly its inertial mass. That is the equivalence principle, the statement that everything falls at the same rate, which the fall that does not depend on what is falling measures to fifteen digits. The absence of dipole gravitational radiation is not a separate fact about gravity. It is the equivalence principle, heard rather than weighed.

And that makes it a test. The binding energy that has to fall too ends with the question of whether a body’s own gravitational binding energy falls like everything else, and points out that a neutron star is a body that is largely its own gravity. If it fell even slightly differently, a binary containing one would carry a moving centre of charge, and it would radiate as a dipole. This essay is about what that radiation would look like, where it would be loudest, and where it has been looked for.

Two charges that cannot radiate a dipole #

Electromagnetism makes the point without any gravity. Two charged bodies in orbit have an electric dipole moment $\mathbf d = q_1\mathbf r_1 + q_2\mathbf r_2$, and a charge that turns must glow says the pair radiates at a rate proportional to $\ddot{\mathbf d}^2$. Now suppose the two bodies have the same ratio of charge to mass, $q_1/m_1 = q_2/m_2 = k$. Then $\mathbf d = k(m_1\mathbf r_1 + m_2\mathbf r_2)$, which is $k$ times the total mass times the centre of mass — and the point that keeps moving as if nothing had happened shows that the centre of mass of an isolated system moves at constant velocity whatever the internal forces do. $\ddot{\mathbf d}$ is zero. The pair radiates no dipole radiation at all, only the far weaker quadrupole.

So “no dipole” does not require the absence of negative charge. It requires that the charge-to-mass ratio be the same for every body in the system. For electric charge that is almost never true — an atom is a proton and an electron with ratios of opposite sign, which is why a charge that turns must glow finds a classical atom collapsing in picoseconds. For gravity it is true by the equivalence principle, and the silence is exact.

The two panels are the same orbit. On the left the centre of charge never moves, whatever the phase; on the right it goes round once per orbit on a small circle, a dipole turning at the orbital frequency. The real mismatch in any viable theory is far smaller than the one drawn, and the size of the circle scales with it, but the difference between a circle of zero radius and a circle of any radius at all is the difference between a system that is silent at dipole order and one that is not.

Where a neutron star could fall differently #

The laboratory measures the equivalence principle for bodies whose gravitational binding energy is negligible. A kilogram of platinum and a kilogram of titanium differ in nuclear binding and in electrostatic energy, and torsion balances find that those forms of energy fall like rest mass to parts in $10^{15}$. Their own gravity contributes a part in $10^{25}$ of their mass, so nothing about how gravitational energy itself falls can be learnt from them.

Theories of gravity other than general relativity generally break the principle in exactly that place, and the oldest of them shows why the place is natural. Nordström’s scalar theory of 1913, the first relativistic theory of gravity, made gravity a single scalar field sourced by the trace of a body’s energy and stress rather than by its energy alone. The orbit special relativity cannot close shows it turning Mercury’s perihelion the wrong way. It also makes a body’s gravitational charge depend on its internal pressure and binding — a photon gas, with zero trace, would have no gravitational charge at all — so the equivalence principle for bound systems fails in it from the start. In scalar–tensor theories — the family that begins with Brans and Dicke, in which a scalar field works alongside the metric — the coupling of a body to the scalar field depends on how much of the body is its own gravitational energy. The dependence is measured by a quantity called the body’s sensitivity, and for a weakly bound body it is close to its fractional binding energy.

The Earth and Moon of lunar laser ranging sit at a few parts in ten thousand million. The Sun is at three parts in a million, a white dwarf at a few parts in a hundred thousand. A neutron star is at thirteen per cent — three hundred million times the Earth — and a black hole, whose mass is entirely field, at the value of one half that the scalar–tensor theories assign it. The estimate drawn is Newtonian, and for the neutron star the true binding needs general relativity and an equation of state; the order of magnitude is what matters, and it is not in doubt. The mass no cold matter can hold up is about exactly how much of such a star’s mass its own gravity can be before nothing holds it up.

This is the ordering the test is after. A pulsar and a white dwarf in orbit are a pair of bodies with vastly different sensitivities: one is thirteen per cent gravity and the other a few thousandths of a per cent. If gravitational binding coupled to anything differently from rest mass, the two would have different gravitational charge-to-mass ratios, and their centre of charge would do what the right-hand panel shows.

A dipole is loudest in a slow orbit #

The two kinds of radiation scale differently with the orbit. With $v$ the relative orbital speed, $\mu$ the reduced mass and $M$ the total, the quadrupole loss of general relativity is

$$L_Q = \tfrac{32}{5}\,\frac{c^5}{G}\left(\frac{\mu}{M}\right)^2\left(\frac{v}{c}\right)^{10},$$

and the loss from a turning scalar dipole, with $\Delta\alpha$ the difference between the two bodies’ couplings to the scalar field, is

$$L_D = \tfrac{1}{3}\,\frac{c^5}{G}\,\Delta\alpha^2\left(\frac{\mu}{M}\right)^2\left(\frac{v}{c}\right)^{8}.$$

The factor of one third is Larmor’s two thirds halved, because a scalar wave has one polarisation where light has two. The difference that matters is in the exponents: the dipole carries one fewer power of $(v/c)^2$. Their ratio is $(5/96)\,\Delta\alpha^2\,(c/v)^2$.

The slower the orbit, the louder a dipole would be relative to what general relativity predicts. That runs against the usual instinct that departures from general relativity will be found where gravity is most violent. For a dipole the opposite holds: in the last seconds of a merger, at a fifth of the speed of light, a mismatch of $\Delta\alpha^2 = 10^{-5}$ would change the loss by a part in ten thousand; in a binary pulsar moving at a thousandth of the speed of light, the same mismatch would add half again to it. Binary pulsars move a million times slower than light in their orbits and they are, for this purpose, the most sensitive instruments there are.

The dipole also arrives at a different frequency. A quadrupole pattern repeats twice per orbit, because a dumbbell looks the same after half a turn, and general relativity’s waves from a circular binary come at twice the orbital frequency. A dipole repeats once per orbit. A detector that saw power at the orbital frequency itself would be seeing something general relativity forbids for any source whose motion is purely orbital — which is why the same principle that tidal forces are the whole of gravity’s local content, argued in the term free fall cannot remove, is also the reason the lowest note of every gravitational wave is the second harmonic of its source.

The same exponents fix the timescales. The orbit that has to shrink computes how the quadrupole loss drags an orbit inwards, with a time to coalescence going as the fourth power of the separation. A dipole loss would add a term going as the third power, dominating early and fading late, and the whole evolution of a wide binary would run faster than general relativity allows.

An 8.5-hour orbit, weighed for a missing loss #

PSR J1738+0333 is a millisecond pulsar of 1.46 solar masses in an orbit of 8.51 hours with a white dwarf of 0.181, found in 2001 and timed for a decade. The white dwarf is bright enough for its own spectrum to be measured, which gives its mass and the mass ratio independently of any theory of gravity, and the orbit is circular to better than a part in a million. Everything general relativity needs to predict the orbital decay is known without using general relativity.

The prediction, $-27.5$ femtoseconds per second, is computed here from the quadrupole luminosity and checked against the textbook closed form; the published value, with the measured masses and their uncertainties, is $-27.7$. The measured orbital decay, after removing the apparent change produced by the system’s motion relative to the Sun and by the Galaxy’s gravitational field, is $-25.9 \pm 3.2$. The two agree. An extra loss of more than about 8 femtoseconds per second — the measured difference plus two standard errors — would have been noticed, and for this pair of masses that bounds the squared difference in scalar coupling below $7.8\times10^{-6}$, a difference in coupling of about three parts in a thousand. The published analysis, with the full treatment of scalar–tensor theories and the uncertainties in the masses, reaches a bound of the same order — comparable to the one from the Cassini spacecraft’s measurement of the Sun’s time delay, and tighter for theories in which strongly bound bodies are especially susceptible to the scalar field, which are exactly the theories a solar-system test cannot reach.

The dashed lines explain why a pair like this was worth waiting for. At periods of minutes the quadrupole loss is steep and large, and a dipole of any plausible size is lost beneath it. At periods of days the dipole lines have closed in on the quadrupole, and a mismatch of $10^{-3}$ would dominate outright. The measurement gets harder at long periods for a mundane reason — the decay becomes too slow to measure in a decade of timing — and the useful systems sit in the middle, at hours, where the loss is still measurable and the dipole has had room to grow.

Why the most famous binary is not the best test #

The Hulse–Taylor pulsar, the first binary pulsar, is the system whose orbital decay first established that gravitational radiation exists. It is not the system that best rules out a dipole, and the reason is its orbit.

An eccentric orbit brings the two stars close together at periastron, where they move fast, and both kinds of radiation are concentrated there. The quadrupole, with its extra powers of the speed, gains more. At the Hulse–Taylor pulsar’s eccentricity of 0.617 the quadrupole loss is almost twelve times what a circular orbit of the same period would lose, while a dipole loss would be only four times larger; any dipole is buried three times deeper under the loss it is being compared with. At an eccentricity of 0.88 the quadrupole gains a factor of 658 and a dipole 57.

So the design of the ideal dipole test is fixed by the two curves and the scaling above: two bodies of very different sensitivities, a circular orbit, and a period long enough for the dipole to have grown but short enough for the decay to be measurable. A millisecond pulsar with a white-dwarf companion in an orbit of hours is exactly that, and it is why pulsar–white-dwarf binaries, not double neutron stars, hold the dipole bounds.

The two tests are complementary rather than redundant. Lunar laser ranging, in the binding energy that has to fall too, weighs the fall of the Earth’s binding energy directly, at a part in $10^{10}$ of the Earth’s mass. The pulsar listens for the radiation that a failure of the same principle would produce, in a body where the binding energy is a tenth of the mass. Both ask whether gravitational energy gravitates like everything else; they ask it at strengths eight orders of magnitude apart.

Where the dipole picture stops #

The scalar–tensor form is one family. The dipole formula here is the one that arises in theories with a scalar field alongside the metric. Theories with an extra vector field, or with a massive graviton, produce dipole radiation with different dependences on the bodies’ properties, and a few theories escape dipole radiation for particular kinds of body. The bound drawn is a bound on a parameter of one family of theories, not on every conceivable departure from general relativity.

The sensitivities are estimates. A neutron star’s sensitivity depends on its equation of state, which is not fully known, and in some scalar–tensor theories a neutron star above a critical compactness can acquire a scalar charge of order one through a nonlinear effect called spontaneous scalarisation. The pulsar bounds are strong precisely because they rule out that effect for the stars observed, but translating a bound on $\Delta\alpha$ into a bound on a theory’s constants requires a model of the star.

The orbits are point masses. Tides, spin and mass loss from the companion all change an orbital period, and the measured decay has to be corrected for the system’s acceleration in the Galaxy, which for J1738+0333 is a correction of the same order as the effect. The two-standard-error allowance used here is a simplification of an analysis that carries all of these.

Black holes are special. A black hole in most scalar–tensor theories carries no scalar charge at all, which gives it the extreme sensitivity drawn in the binding figure. A black hole with a neutron-star companion would therefore make a strong dipole emitter if the neutron star carried a charge. Such systems have been seen merging in gravitational waves; none has yet been found as a pulsar in a wide orbit.

The scalar wave no drawing of the orbit shows #

The first figure shows the centre of charge circling and cannot show the field that circling produces: a scalar wave at the orbital frequency, half the frequency of the quadrupole wave, spreading with a different pattern on the sky and exciting a detector differently from either polarisation of general relativity’s waves. The period-decay figure places one measured point on a line of predictions; it cannot show the decade of timing residuals behind it, the corrections for Galactic acceleration that are comparable to the effect, or the optical spectroscopy of the white dwarf that fixes the masses. And the binding figure is a Newtonian estimate for bodies at least one of which is emphatically not Newtonian. What the figures establish is the scaling: which systems a dipole would be loudest in, and why the circular, hours-long pulsar–white-dwarf pair is the instrument for it.

Still open: whether a strong field hides what a weak one shows #

Binary pulsars test the principle where each body is strongly bound but the orbit is slow and wide. Merging black holes and neutron stars test gravity where the orbit itself is strongly relativistic, and there a dipole is relatively quiet but other departures from general relativity — in how the waves propagate, in how a merged object rings — are loud. Some theories have been built to pass every weak-field test by suppressing their extra field in dense environments, and whether any of them can survive both the pulsar bounds and the merger observations at once is being worked out as the catalogue of events grows. A black hole orbiting a pulsar, if one is found, would close the gap between the two regimes in a single system.

The habit worth carrying away is to ask what a missing term is missing because of. A radiation law that starts at a higher order than it might is usually a conservation law or a symmetry in disguise, and a violation of that symmetry announces itself first as the lower order coming back. Gravity’s missing dipole is the equivalence principle; its return would be the first sign that a neutron star’s own weight falls differently, and it would be heard most clearly not in a merger, but in an orbit so slow it takes eight hours to go round.